A Novel Analysis Method for Emission Tomography

Costas N. Papanicolas,1, ∗ Loizos Koutsantonis,1 and Efstathios Stiliaris2, 1

1The Cyprus Institute, Konstantinou Kavafi 20, 2121 Nicosia, Cyprus2Physics Department, National and Kapodistrian University of Athens,

Ilissia University Campus, 15771 Athens, Greece(Dated: June 15, 2021)

We present a novel analysis method for image reconstruction in emission tomography. Themethod, named Reconstructed Image from Simulations Ensemble (RISE), utilizes statistical physicsconcepts and Monte Carlo techniques to extract the parameters of a physical model representingthe imaged object from its planar projections. Its capabilities are demonstrated and evaluated byreconstructing tomographic images from sets of simulated SPECT projections. The RISE resultscompare favourably to those derived from the well - known Maximum Likelihood Expectation Max-imization (MLEM) method, the Algebraic Reconstruction Technique (ART) and the Filtered BackProjection (FBP).

I. INTRODUCTION

Emission tomography is playing a dominant role inmedical imaging with Positron Emission Tomography(PET) [1–4] being the predominant modality followed bythe Single Photon Emission Computerized Tomography(SPECT) [5–8]. Other modalities such as Infrared Emis-sion Tomography (IRET), relying on the same generalprinciples are currently being investigated [9].

In all emission tomography modalities the quality ofthe reconstructed images is limited by the backgroundradiation emerging from the surrounding medium (e.g.tissue), absorption and re-scattering effects [10–15]. Inthe case of PET and SPECT where the use of radiophar-maceuticals is required, and which are detrimental to thehealth of the patient, it is desired to achieve good imagereconstruction while keeping the injected dose to a min-imum. Minimizing the dose implies reconstruction fromfewer images and/or with limited statistics.

In an effort to address the aforementioned considera-tions a novel method has been developed in which the2D or 3D tomographic images are reconstructed from anensemble of simulated solutions by statistically weigh-ing the ones that satisfy a ”goodness” criterion to theobserved data. This method, in the interest of brevity,will be referred below as ”Reconstructed Image fromSimulations Ensemble” (RISE).

RISE and the underlying Athens Model IndependentAnalysis Scheme (AMIAS) method [16] provide a highlycomplex and computationally intensive method for in-verse problems which has proven successful in quantuminverse problems (scattering) in nuclear [17–19] and par-ticle physics [20]. It is based on statistical physics con-cepts and its implementation relies on Monte Carlo simu-lation techniques. It is a method of general applicability,well-suited for cases characterized by low statistics, noisyand attenuated data, as is often the case for SPECT andPET.

∗ Corresponding author: papanicolas@cyi.ac.cy

We present in subsequent sections the theoreticalframework describing RISE and its computational im-plementation. We present its features in concrete ex-amples in the case of SPECT modality. Software phan-toms are employed to test the capabilities of the methodand benchmark it versus the well established and well-understood techniques, the Maximum Likelihood Ex-pectation Maximization (MLEM), the Algebraic Recon-struction Technique (ART) and the Filtered Back Pro-jection (FBP).

II. METHODOLOGY

A. The Tomographic Problem

In emission tomography, the sectional image of an ob-ject from which radiation is emitted is reconstructed fromprojection measurements {Yi| i = 1 · · ·NP × NR} ob-tained at different angles. NP is the total number ofprojection angles and NR is the number of bin (pixe-lated) measurements per projection angle.

The prevailing practice in the field of medical imagingis to represent the N × N pixelated tomographic imageby the vector {Fj |j = 1 · · ·N×N}. The relation betweenthe set of measured quantities {Yi} to the set of elements{Fj} can be expressed as:

Yi =

NP×NR∑j=1

PijFj (1)

where Pij is a weighting matrix, also referred to as theprojection matrix [21] linking the image vector Fj to theset of projections Yi. Fj denotes the intensity of theemitted radiation from the elemental area defined by thejth pixel.

In the implementation of RISE presented in this pa-per, we do not examine issues of attenuation and scat-tering. The various approaches dealing with these issuescan easily be implemented in RISE and indeed be furtherextended.

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B. Established Reconstruction Techniques

The commonly used image reconstruction techniquesin emission tomography are classified into two main cat-egories: The first includes analytical methods such as theFiltered Back Projection (FBP) and its variants. Thesetechniques, based on analytic inversion formulae [22, 23]of the Radon transform [24] and utilizing additional con-volution kernels act on either the projection or imagespace, to provide images requiring minimal computa-tional effort. The second category comprises the iterativemethods that approximate the tomographic problem asa system of linear equations [25, 26]. These methodsproduce the reconstructed image following an iterativescheme for the minimization of a predefined cost functionor maximization of the likelihood function. The mostwidely used iterative methods in emission tomographyare the Algebraic based Reconstruction Technique (ART)[27, 28] and the the Maximum Likelihood ExpectationMaximization (MLEM) [29, 30] including its acceleratedversion, the Ordered Subsets Expectation Maximization(OSEM) method [31]. Compared to the Filtered BackProjection (FBP), the iterative methods exhibit slowerconvergence but lead to more reliable reconstruction re-sults especially in tomographic problems where a limitednumber of planar projections are available [32–34] and/orthese projections are characterized by noise and attenu-ation.

In this study, we employed the traditional FBP, theNewton-Raphson version of ART [35] and an open-sourceversion of MLEM [36] to produce reconstructions forcomparison to the images produced by RISE.

C. The Reconstructed Image from SimulationsEnsemble (RISE) Technique

The Reconstructed Image from Simulations Ensemble(RISE) technique approaches the tomographic problemin an entirely different way which is based on the AthensModel Independent Analysis Scheme (AMIAS) method-ology. RISE obtains the image of the object (in 3D) ora 2D sectional image of it by constructing a model rep-resenting the radiation emission sources in the imagedvolume. The model can be very general so as not to biasthe outcome or specific and restrictive if it is desirable toincorporate prior knowledge. The model chosen to rep-resent the imaged object is characterized by a numberof parameters which are determined from the imagingdata. In principle, the best model representation canbe obtained from the data utilizing any goodness of fitmethod; we opted to employ the rather elaborate formu-lation of AMIAS which was formulated and successfullyemployed in addressing similar (inverse) problems.

1. The Athens Model Independent Analysis Scheme(AMIAS)

Methods based on AMIAS are applicable in problemsin which the parameters of interest (e. g. the locationand size of radiating ”hotspots”) are linked to the observ-able quantities (data) through a model. They incorpo-rate Monte Carlo techniques to simulate the generation ofthe observable quantities and as such, they require heavycomputational resources. Prior knowledge can be incor-porated endowing the scheme with Bayesian capabilities.In order to extract the desired parameter values fromthe data, the method employs an appropriate ”goodness”criterion to quantify the comparison between the modelsimulated predictions to the observables; typically this ischosen to be the χ2 criterion:

χ2 =∑i

(Yi − Yi)2

ε2i(2)

where Yi is the predicted by the model quantity, Yi is thecorresponding measured quantity and εi is its associateduncertainty.

The method extracts the model parameters values andthe corresponding uncertainties from a set of measuredquantities (observables). The extracted parameters allowthe reconstruction of the imaged object according to themodel employed to represent the data.

2. RISE: Implementing AMIAS in emission tomography

In AMIAS the tomographic imaging problem is ap-proached in the following manner: the plane or volumeto be imaged is described by a model whose parametersare derived via the AMIAS methodology. Unless priorknowledge is to be incorporated, It is necessary that themodel has adequate flexibility to accommodate all shapesand intensity variations, that could be expected, and thatit introduces no model bias. As AMIAS derives the pa-rameters of the model with the maximum precision thedata allow and with an evaluation of their uncertainties,it offers the added benefit of attributing a level of confi-dence to the parametrization of the derived result.

In RISE the AMIAS methodology is implemented asfollows (Figure 1):

1. A model is adopted in which the imaged distribu-tion of the radiation sources is parameterized. Inthe variant presented here, the radiation distribu-tion is described by a sum of elementary shapes rep-resenting the hotspots and a sum of smoothly andslowly varying terms representing the background.Each elementary shape is specified by a set of pa-rameters (e.g. position, size, intensity etc) and eachbackground term by its amplitude.

3

FIG. 1. Flowchart of the algorithmic implementation of theAMIAS methodology[16] in emission tomography. The pa-rameters of a model describing the sources of radiation arerandomly sampled in an iterative scheme. The simulated dataare compared to the measured data to quantify the goodnessof the simulated solution. From a sufficiently large ensembleof solutions, numerical results are extracted for the param-eters best describing the image by examining the statisticalproperties of the ensemble.

2. A random sampling procedure is used to choose aset of parameter values in a predefined range. Theset of randomly chosen parameters values uniquelyand completely define the sources of radiation andas such, they provide a possible ’solution’ to theproblem; each set of model parameters correspondsto a given image reconstruction. In the choice ofthe model parametrization and of the ranges forthe sampling of model parameters prior knowledgecan easily be incorporated.

3. The tomographic image of the ’object’ is con-structed from the randomly chosen values of pa-rameters. Projections of the ’object’ are computedto simulate the measurement process incorporatingthe exact geometry of the imaging set-up and otherprocesses such as the attenuation and scattering ofphotons in the intervening medium.

4. The simulated projections are compared to themeasured projections by using the χ2 criterion (Eq:2). The χ2 value is assigned to each ’solution’ toquantify its ’goodness’ of representing reality, ascaptured in the experimental dataset.

5. Steps 2−4 are repeated (typically 104−106 times)

in order to construct a large ensemble of solutions.

6. Numerical results for the parameters best describ-ing the data (Probability Distribution Functions,mean values, and uncertainties) are extracted byexamining the statistical properties of the ensem-ble. The derived parameters allow the reconstruc-tion of the model represantation of the imagedobject.

7. The intensity of radiation from each individualpixel (voxel) of the model representation is allowedto vary to further improve the agreement of the re-constructed image with the data. This result in theRISE image reconstruction .

3. Modeling the Imaged Object

In the case studies presented here, and in imple-menting the first step of the RISE algorithm (a modelparametrization of the imaged object) the 3D distribu-tion of radiation sources is represented by a stack of 2Dtomographic images, each one associated to a differentoffset in the third (”vertical”) axis. The geometry of theactivity distribution to be imaged in the 2D space is ap-proximated by a set of fundamental shapes. In this work,these shapes are chosen to be ellipses which imply ellip-soidal shapes of radiation emitters in 3D. The parametricequation of an ellipse is given in the form of:

R(θ) =u · v√

(v · cos(θ − φ))2 + (u · sin(θ − φ))2(3)

where u and v are the semi-major and semi-minor axesof the source (”hotspot”) respectively and φ is the angledetermining the orientation of the ellipse in the tomo-graphic plane. The meaning of each parameter is bestunderstood with the help of Figure 2. The intensity ofthe radiation emitted by each point (pixel) in the volumespecified defined by this elementary shape is described byan intensity profile function.

The activity distribution in the imaged tomographicsection is modeled by a sum of these elementary ellipses,each one (or a cluster of them) representing the presenceof an emitting source (”hotspot”). Three different inten-sity profile functions providing the intensity distributionof the sources as a function of the geometrical factor Rare examined in this study. All of them assume onlyradial dependence.

The first model (”Gaussian”) assumes a Gaussian dis-tribution:

T (x, y) =

N∑k=1

Ak · exp(−1

2

r2kR2

k

) (4)

where, Ak is the activity at the center of the kth

”hotspot”, N is the number of ”hotspots” and rk isthe euclidean distance between the image element repre-sented by its physical coordinates (x, y) and the centroid

4

FIG. 2. Left panel: Geometric representation of the 2D activity distribution T (x, y) modeled according to Equation 9; thegeometric parameters in Equation 3 describing the imaged object are depicted in the figure. The gaussian distribution (Equation4) was used to describe the activity intensity profile in each hotspot (indicated by H1-H3). Right panel: The radial profile ofthe three distribution functions used in modeling the profile of the radionuclide concentration in this paper.

(xk, yk) of the kth ”hotspot” (schematic illustration pro-vided in Figure 2). T (x, y) is the activity value of thetomographic image element.

In the second model (”Fermi”), the distribution of ra-dioactivity is represented by the Fermi-like function:

T (x, y) =

N∑k=1

Ak ·(exp(

rk −Rk

skRk) + 1

)−1

(5)

where, in this model, an additional parameter sk (di-fussnes) is employed.

The third model (”step”) assumes a uniform activitydistribution within the geometrical confines of the im-aged hotspot and an abrupt change at its geometricaledges:

T (x, y) =

N∑k=1

Ak ·Θ(Rk − rk) (6)

where:

Θ(Rk − rk) =

{1, for Rk ≥ rk0, for Rk < rk

The surrounding medium is also assumed to be anemitter of radiation having a simple slow varying spa-tial distribution:

B(x, y) =

M∑i=0

CiYi(x, y) (7)

where Yi(x, y) is a set of basis functions, Ci are their am-plitudes and M is the number of radial terms chosen todescribe the background distribution. In this study wedescribe the radiation emitted by the medium (”back-

ground”) by using the set of Zernike polynomials Zji (x, y)

[37, 38]. Thus:

B(x, y) =

Mz∑i=0

i∑j=−i

Cji Z

ji (x, y) (8)

Thus the 2D imaged object is assumed to have an activitydistribution F (x, y) given by:

F (x, y) = T (x, y) +B(x, y) (9)

4. Determining the model parameters

The RISE reconstruction process implements the al-gorithmic framework shown in Fig. 1. It allows the de-termination of an adequate number of background terms(M) and the number of ”hotspots” (N) which are neededto represent the imaged distribution (see Equation 9). Itresults in the extraction of numerical values and associ-ated uncertainties for the parameters of the backgroundand for each one of the N modeled ”hotspots”.

RISE employs a fixed number of terms (M) in Equa-tion 7 to represent the background distribution. Highspatial frequency polynomials are excluded to avoid

5

FIG. 3. Upper panel: A) Software ”Phantom A”, the image of activity distribution simulated by a set of Gaussian functions(same as Figure 2). B) The derived image providing the model representation of the phantom in which the parameters ofinterest are determined from their Probability Distribution Functions (PDF), tabulated in Table I. C) The RISE reconstructedimage of the phantom derived from the model representation using the last step of the RISE algorithm. Bottom panel: PDFsderived by RISE determining the parameters of the H3 ”hotspot” in ”Phantom A”. Mean values and uncertainties calculatedfrom the PDFs are shown at the top-left of the plots; the generator values used to simulate the phantom activity distributionare indicated by the solid lines. Values are shown in units of pixel-widths.

the spatial frequencies corresponding to the targeted”hotspots”. Thus, the choice of a reasonably low num-ber M prevents the correlation between the coefficientsdescribing the higher frequencies of background B(x, y)and the parameters describing the target T (x, y).

A number of algorithms were used to determine N .The easiest to explain is the one in which N is deter-mined by sequentially increasing the number of terms(N = N + 1) starting from N = 0 till convergence isreached. For each increment of N , the Bayesian Infor-mation Criterion (BIC) [39, 40] is monitored and used to

quantify the ”goodness” of the model of N terms:

BIC = χ2min(N) + (n ·N +M) · log(NP ·NR) (10)

where χ2min(N) is the minimum χ2 value in the ensemble

of solutions constructed for the model of N terms, n isthe number of parameters describing each ”hotspot”, and(NP ·NR) is the length of the sinogram. The optimumnumber of terms N is selected as the one yielding theminimum BIC score which defines ”convergence”.

Having chosen the optimum number of terms (N) inEquation 9 the corresponding ensemble of solutions isused to determine the values of parameters for the modelselected to describe the data. This ensemble of solutions

6

TABLE I. Parameters and associated uncertainties derived in RISE to produce the model representation of the phantom shownin Figure 2. The ”true” values used to describe the three hotspots H1, H2 and H3 and the background (B) are also shown.The empirical values and associated uncertainties for H3 are those derived from the PDFs shown in Figure 3.

A x y u v φ

H1 1.0 40.0 90.0 3.0 3.0 0.0

1.1 ± 0.3 40.7 ± 0.9 89.8 ± 0.7 3.5 ± 0.9 3.5 ± 0.8 0.1 ± 0.8

H2 1.0 55.0 90.0 3.0 3.0 0.0

0.9 ± 0.3 55.4 ± 1.2 89.4 ± 0.9 4.0 ± 1.7 3.7 ± 1.5 0.0 ± 0.8

H3 1.0 88.0 54.0 12.0 3.0 0.35

0.9 ± 0.1 88.2 ± 0.6 54.7 ± 1.2 12.9 ± 1.7 3.6 ± 0.7 0.37 ± 0.16

C00 C1

1 C−11 C−2

2 C02 C2

2

B 0.6 0.08 0.0 0.04 0.0 0.0

0.597 ± 0.004 0.07 ± 0.01 0.005 ± 0.007 0.033 ± 0.007 0.003 ± 0.008 0.001 ± 0.009

is then used to derive the Probability Distribution Func-tion (PDF) of each one of the model parameters as spec-ified by the AMIAS methodology [16]. Mean values andhigher-order moments are extracted from PDFs whichprovide the parameters’ conventional optimum values.

5. Illustrative Case

To illustrate some key features of the method we applythe RISE methodology to reconstruct the 128× 128 im-age of ”Phantom A”, shown in Figure 2 and panel A ofFigure 3. The phantom used comprises three ”hotspots”,two circulars in shape (H1 and H2) and a larger of ellip-soidal shape (H3); all three were given gaussian intensityprofile. The background was described by the sum ofthe first six Zernike polynomials. Simulated data (sino-grams) for 24 equidistant projections were generated us-ing Equation 1 which were subsequently randomized withPoisson noise. A total of 24 parameters were used to de-scribe the phantom.

In applying the algorithm described in Section II C 2,we modeled the object to be imaged by Equation 4 as-suming a Gaussian intensity profile. Following the pro-cedure of Section II C 4 it was found that the phantomis best described by three (N = 3) hotspots; the back-ground was assumed to be adequately described by thesum of the first ten Zernike Polynomials (M = 10,Mz =3). An ensemble of solutions was constructed compris-ing of 15000 simulated solutions. Employing the AMIASmethodology PDFs for each of the 28 free parametersof the model were derived in the 6th step of the RISEalgorithm which allows the model reconstruction of thephantom shown in panel B of the Figure 3.

The derived PDFs for the position (x, y), size (u, v),orientation (φ) and intensity (A) of hotspot H3 are shown

in Figure 3, along with the corresponding generator val-ues. The values of the generator parameters and thoseextracted by RISE are shown in Table I, for all 24 in-put parameters of the model. Excellent agreement isobserved between the derived and generator parametervalues.

The model representation is subsequently varied asprescribed in the 7th step of the RISE algorithm to yieldthe reconstructed tomographic image which is shown inpanel C of the same figure.

III. SIMULATION STUDIES ANDEVALUATION OF RISE

Software phantom simulations allowing direct similar-ity measures between the reconstructed images and theknown true distribution were performed to evaluate anddemonstrate the capabilities of the RISE method. Phan-toms simulating the distribution of radiation sources wereconstructed on 2D grids and projection data were pro-duced by using the forward projection model presentedin Equation 1.

A. Similarity Measures

The assessment of the efficacy of a given reconstructionalgorithm benefits greatly by the use of a quantitativecriterion. For software phantoms the procedure is simple:the reconstructed distribution is compared to the known’true’ distribution by employing a merit function. Threewell understood and widely used metrics, the NormalizedMean Square Error (NMSE), the Correlation Coefficient(CC) and the Peak Signal to Noise Ratio (PSNR) wereused.

7

The Normalized Mean Square Error (NMSE) providesa measure of the overall reconstruction error:

NMSE =

∑N2

i=1(F ri − F t

i )2∑N2

i=1 Fti2

(11)

where F ri , F t

i are the pixel values, N2 of them, of thereconstructed and phantom (”true”) image respectively.NMSE providing a normalized measure of the absolutedifferences between the two images can be used in com-parison studies exhibiting different activity ranges.

The Correlation Coefficient (CC) allows similaritymeasures between two images without requiring thequantification of their absolute difference [34, 41]. CCis defined as:

CC ≡∑N2

i=1(F ri − F r)(F t

i − F t)√∑N2

i=1(F ri − F r)2

∑N2

i=1(F ti − F t)2

(12)

where F r and F t are the mean values of the image ele-ments F r

i and F ti respectively. Depending on the relation

between the two images, reconstructed and phantom, thecorrelation coefficient attains values between 0.0 and 1.0.

The Peak Signal-to-Noise Ratio (PSNR) measured indecibels, is calculated by:

PSNR = 10 log10

(N2 ·max(F r)∑N2

i=1(F ri − F t

i )2

)(13)

The higher the PSNR, the better the quality of the re-constructed image.

B. Software Phantoms

Two software phantoms were used to evaluate anddemonstrate the capabilities of the RISE method:

• ”Phantom A” shown in Figure 2 is comprised ofthree hotspots of ellipsoidal shape exhibiting Gaus-sian intensity profile immersed in a non-uniformslow varying background activity.

• ”Phantom B” shown in Figure 4, is a complexstructure, an anagram, an overlay of the threeGreek letters (P , Π, K) in a continuous uniformdistribution with sharp (step) edges, placed in zerobackground.

Although the methodology presented and the examplesdiscussed are of general applicability to all modalities ofemission tomography, the SPECT modality allows theclosest realization of the simulation studies presented.

For each phantom, the image of the ”true” activitydistribution was sampled on a rectangular grid of 128× 128 pixels size. Sets of vectorized projections (sino-grams) were generated from the ”true” images throughEquation 1 by simulating 24 projections, evenly spaced in

FIG. 4. Software ”Phantom B”, a complex anagram withsharp edges, used in simulation studies to validate the RISEmethod.

the full (360o) angular range. The generated projections{Y t

i , i = 1 . . . NP × NR} obtained from the phantomimages were further randomized with a Poisson probabil-ity distribution to provide the noisy sets of projections{Yi, i = 1 . . . NP ×NR}:

Yi ∼ Pois(Y ti ) (14)

The phantom pseudodata was used to examine threedifferent aspects of the RISE method:

A Model Dependence: To showcase the ability of RISEto reconstruct adequately the ”true” activity dis-tribution even in cases where the modeling of theelemental hotspots are only approximate. It isachieved by examining the dependence of the re-construction on the choice of the intensity profilefunction for the targeted hotspots (Gaussian, Fermiand Step Function). Images of the ”Phantom A”were reconstructed and compared.

B Shape Flexibility: To demonstrate the ability ofRISE to reproduce any arbitrary shape especiallyin cases where the elemental hotspots employeddo not resemble the geometrical characteristics ofthe imaged object. In this case study the imageof ”Phantom B” was reconstructed from its noisyprojections by employing the series of ellipsoidallyshaped sources.

C Minimization of Exposure The ability of RISE toextract the ”true” distribution from reduced statis-tics (in nuclear medical imaging corresponding tominimizing the exposure of the patient to radia-tion) is showcased. In a SPECT imaging simula-tion, ”Phantom A” was used in three simulation

8

FIG. 5. Images of ”Phantom A” reconstructed with the RISE method to evaluate the dependence of the reconstruction on thechoice of the model that is used to represent the targeted hotspots in the tomographic plane. Three different models were used,the Gaussian (M1), the Fermi(M2) and the Step Function (M3).

cases (D1, D2, D3) to produce three sets of 24 pro-jections. The data were generated by simulating8048 (D1: full dose), 4028 (D2: half dose) and 2014(D3: quarter dose) photon counts respectively andrandomized with Poisson noise. Reconstructed im-ages were obtained by using the Fermi model in theframework of RISE.

Reconstructions providing reference images for com-parison were obtained in cases studies B and C withMLEM, ART, and FBP.

IV. RECONSTRUCTION RESULTS

A. Model Dependence

Figure 5 shows the reconstructed images of ”PhantomA” as obtained in the first case study. The images werereconstructed by using the three intensity profile func-tions: Gaussian, Fermi and step defined by Equations 4,5 and 6 respectively.

A casual visual inspection of the reconstructed imagesshown in Figure 5 reveals that the three images are verysimilar even in the case of the unrealistic step functiondistribution (M3). This is borne out by the compari-son of the corresponding CC, NMSE and PSNR valuesshown in Table II. The Gaussian model was expected toyield the best agreement as the same model was used togenerate the ”true” image of the phantom. Neverthe-less, similar CC, NMSE and PSNR values to those of theGaussian were obtained by using the Fermi model. Thelowest values were obtained for the step model. The stepfunction was chosen to examine the ability of RISE to ac-commodate a manifestly deficient model which can notoffer an acceptable representation of the phantom distri-bution. In such deficient model representation, the last

TABLE II. CC, NMSE and PSNR values comparing themodels used in RISE to reconstruct the image of ”PhantomA”. Values for both the model representation and the recon-structed (R) images are given.

Model CC NMSE PSNR

Gaussian 0.94 0.029 24.86

Gaussian (R) 0.98 0.012 28.65

Fermi 0.94 0.029 24.10

Fermi (R) 0.98 0.013 27.87

Step 0.93 0.035 23.22

Step (R) 0.97 0.017 26.27

step of the RISE algorithm proves to be important forthe correction of the image. In the case of step functionmodeling, the CC value characterizing the reconstructedimage was 1% lower than that obtained with the Gaus-sian model. It is demonstrated that the choice of a modelfor representing the intensity distribution is not criticalfor arriving at the correct outcome.

B. Shape Approximation

RISE was applied to reconstruct the image of ”Phan-tom B” by employing the summation of elementary el-lipses having a Fermi activity profile as the approximat-ing model. RISE model representation and reconstruc-tion (A1 and A2 respectively) images are shown in Fig-ure 6. Images reconstructed with MLEM (B1), ART (C1)and FBP (D1) are also shown for comparison. In the caseof MLEM and ART, seven and two grand iterations were

9

FIG. 6. Top row: The RISE, model representation (A1) and reconstructed (A2) images of ”Phantom B” and the unfilteredimages provided by MLEM (B1), ART (C1) and FBP (D1). Bottom row: The MLEM (B2), ART (C2) and FBP (D2) imagespost-filtered with a low-pass Butterworth filter are also shown.

performed respectively. MLEM, ART and FBP recon-structions were further post-filtered with a 3rd order But-terworth filter of 0.25 cycles per pixel cut-off frequencyand are shown in panels B2, C2 and D2 respectively.

The reconstructed images of ”Phantom B” were eval-uated in terms of CC, NMSE, and PSNR; the results areshown in Table III. Three of the four methods (RISE,MLEM, ART) yield comparable CC, MSE and PSNRvalues. The result indicates that, in cases where the truedistribution is totally unknown and has a complex struc-ture, the choice of the RISE method leads to similar re-constructions to those of MLEM and ART. FBP, affectedby the angular undersampling (24 projections were used),produced images of lower CC and CNR values as com-pared to the other three methods.

C. Minimization of Exposure

Figure 7 shows the reconstructed images of ”PhantomA” as obtained with RISE, MLEM, ART, and FBP forthe three simulation cases varying the number of simu-lated photons’ counts (D1:8048 counts, D2:4024 counts,D3:2012 counts). RISE reconstruction was performed byemploying the Fermi model. MLEM, ART and FBP im-ages were reconstructed and post-filtered as described inSection IV B.

A visual examination and comparison of the resultsreveal that the RISE image exhibits higher contrast andincreased detectability of hotspots compared to those of

TABLE III. CC, NMSE and PSNR values evaluating the re-constructions of ”Phantom B” obtained from RISE, MLEM,ART and FBP. The evaluated image as shown in Figure 6 isdenoted in column ”Panel”.

Panel CC NMSE PSNR

RISE A1 0.93 0.087 17.31

RISE (R) A2 0.94 0.070 18.63

MLEM B1 0.92 0.093 21.08

MLEM (PF) B2 0.94 0.072 20.56

ART C1 0.92 0.105 18.90

ART (PF) C2 0.93 0.085 18.64

FBP D1 0.88 0.254 17.78

FBP (PF) D2 0.89 0.234 17.74

MLEM. Compared to the images obtained with ART andFBP, RISE images present less amount of noisy artifacts.

CC, PSNR and NMSE values evaluating the recon-structed images are shown in Table IV. In all simula-tion cases, RISE image scored the highest CC values fol-lowed by the post-filtered MLEM images. Furthermore,RISE led to an improvement in PSNR as compared tothe PSNR values of the three other methods.

10

FIG. 7. Reconstructed images of ”Phantom A” obtained from the simulated data in the three cases characterized by diminishingdoses (D1 = 2D2 = 4D3). The presented MLEM, ART and FBP image were post-filtered with a low-pass filter. The presentedimages were scaled to their activity ranges in order to be visualized with the maximum possible contrast.

TABLE IV. CC, NMSE and PSNR values calculated from the reconstructed images in the three simulation cases varying thenumber of simulated photons (images are shown in Figure 7).

D1 (Full Dose) D2 (Half Dose) D3 (Quarter Dose)

CC NMSE PSNR CC NMSE PSNR CC NMSE PSNR

RISE 0.98 0.013 27.87 0.97 0.017 26.17 0.92 0.041 21.28

MLEM (PF) 0.94 0.035 23.82 0.90 0.050 19.61 0.86 0.068 18.77

ART (PF) 0.90 0.055 21.15 0.80 0.127 18.34 0.71 0.203 17.80

FBP (PF) 0.80 0.402 16.87 0.75 0.433 15.77 0.73 0.497 15.47

V. SPECIFIC APPLICATIONS

Early results from on-going work applying the theoret-ical framework (RISE) presented in this paper have been

recently reported for the evaluation and demonstrationof the method in purpose-specific applications includingexperimentation with hardware phantoms. These studiesinclude SPECT Dopamine Transporter Imaging (DAT)for Parkinson’s disease [42], imaging of lungs and kidneys

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in mice [43] and extensive experimentation with 99mTcphantoms utilizing the high-resolution SPECT camera ofthe University of Athens [44]. Moreover, the employmentof the method in other modalities of emission tomogra-phy such as the Infrared Emission Tomography (IRET) iscurrently under evaluation. Results from an initial inves-tigation of the RISE method in IRET, with attenuatedand diffussed thermal data, can be found in a preliminarystudy[45].

VI. SUMMARY CONCLUSIONS

The theoretical foundation of a novel method, the Re-constructed Image from Simulations Ensemble (RISE),for emission tomography, has been presented along withits algorithmic implementation. In the work presentedhere, RISE was demonstrated and evaluated with soft-ware phantoms in the SPECT modality resulting supe-rior reconstruction results compared to the well estab-lished MLEM, ART, and FBP methods.

The method is computationally demanding but robustand particularly well suited to noisy and low statisticsdata. The new method allows quality reconstructed datafrom fewer planar images and with lower statistics, thusallowing the use of a lower dose of a radiopharmaceuticalin SPECT and presumably PET imaging, minimizing the

exposure to the patient.The RISE method, as presented here, provides a recon-

struction framework of 2D tomographic images. The ex-tension of the method to a 3D model providing the directreconstruction of the volumetric image is straightforwardand it is expected to provide further improvement to the2D images; early results from the SPECT tomographicimaging of hardware phantoms appear to validate thisexpectation.

ACKNOWLEDGMENTS

This work was supported by the Graduate School ofThe Cyprus Institute and the Cy-Tera Project ”NEAIPODOMI/STRATI/0308/31”, which is co-funded bythe European Regional Development Fund and the Re-public of Cyprus through the Research Promotion Foun-dation. The authors would like to thank TheodorosChristoudias and Charalambos Chrysostomou for proof-reading this manuscript.

CONFLICTS OF INTEREST

The authors have no conflicts of interest, financial orother, to declare.

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